College Algebra: Section 1.6Equations and Inequalities Involving Absolute Value

Objectives of this Section

• Solve Equations Involving Absolute Value

• Solve Inequalities Involving Absolute Value

Equations Involving Absolute Value

If the absolute value of an expression equals some positive number a, then the expression itself equals either a or -a. Thus,

u a u a u a i s e q u i v a l e n t t o o r

Solve: 2 3 11x

2 3 11x

2 3 11 2 3 11x x or 2 14x x 7

2 8x x 4

Solution set: {-4, 7}

u a a u a

u a a u a

i s e q u i v a l e n t t o

i s e q u i v a l e n t t o

Theorem

In other words, |u| < a is equivalentto -a < u and u < a.

Ex. Solve y + 21 ≥ 7 - 21 -21

y ≥ -14

(-14) + 21 ≥ 7

7 ≥ 7

1. Draw the “river”2. Subtract 21 from

both sides3. Simplify4. Check your answer5. Graph the solution

-14 -13-15●

Ex. Solve 8y + 3 > 9y - 14

o17 1816

- 8y - 8y

3 > y - 14

+ 14 + 14

17 > y

y < 178(16) + 3 > 9(16) – 14

131 > 130

1. Draw “the river”

2. Subtract 8y from both sides

3. Simplify

4. Add 14 to both sides

5. Simplify

6. Rewrite inequality with the variable first

7. Check your answer

8. Graph the solution

Ex. Solve 8y + 3 > 9y - 14

o17 1816

- 8y - 8y

3 > y - 14

+ 14 + 14

17 > y

y < 178(16) + 3 = 9(16) – 14

131 > 130

Big Tip!!! At the end of solving inequality, always put the variable at the LEFT hand side. Then arrow of the inequality sign tells you the correct graph

y < 17The graph should toward to the left.

To Solve the Absolute Value Inequalities

1. Isolate the absolute value expression.2. Make sure the absolute value inequality can be defined.3. For any defined absolute value inequality with template:

|X| a, where a > 0 and = <, >, ≤, ≥ write the corresponding compound inequalities by

following the rules:

a) if “|X| > a” or “|X| ≥ a”, meaning “greator”, “leaving the jail”

b) set “jail boundaries” as “–a” and “a”c) write compound inequalities as

X < –a or X > a meaning “stay left to the left boundary or right to the right boundary”

d) if “|X| < a” or “|X| ≤ a”, meaning “less thand”, “going to the jail” e) set “jail boundaries” as “–a” and “a”f) write compound inequalities as

–a < X < a (and) meaning “stay between the two boundaries”

4. Solve the converted compound inequalities.

Solve | x – 3 | < 5

Absolute Value Inequality

(1)The absolute value expression is isolated(2)It is a well defined absolute value inequality(3)It is a “less thand” inequality. (go to the jail)

a) set jail boundaries: –5, and 5b) write compound inequalities: (stay in

between the jail boundaries)

–5 < x – 3 < 5

–2 < x < 8

+3 +3 +3

You try this!Solve | x + 4 | < 1

Absolute Value Inequality

(1)The absolute value expression is isolated(2)It is a well defined absolute value inequality(3)It is a “less thand” inequality. (go to the jail)

a) set jail boundaries: –1, and 1b) write compound inequalities: (stay in

between the jail boundaries)

–1 < x + 4 < 1

–5 < x < –3

– 4 – 4 – 4

Solve | 2x + 3 | – 3 ≥ 2

Absolute Value Inequality

(1)The absolute value expression is NOT isolated| 2x + 3 | – 3 ≥ 2

| 2x + 3 | ≥ 5(2)It is a well defined absolute value inequality(3)It is a “greator” inequality. (leaving the jail)

a) set jail boundaries: –5, and 5b) write compound inequalities: (stay left to left

boundary and right to the right boundary)

2x + 3 ≤ –5 or 2x + 3 ≥ 5

2x ≤ –8 or 2x ≥ 2 x ≤ –4 or x ≥ 1

– 3 – 3

+ 3 + 3

– 3 – 3

You try this! Solve | 4x – 3 | + 5 ≥ 8

Absolute Value Inequality

(1)The absolute value expression is NOT isolated| 4x – 3| + 5 ≥ 8

| 4x – 3 ≥ 3(2)It is a well defined absolute value inequality(3)It is a “greator” inequality. (leaving the jail)

a) set jail boundaries: –3, and 3b) write compound inequalities: (stay left to left

boundary and right to the right boundary)

4x – 3 ≤ –3 or 4x – 3 ≥ 3

4x ≤ 0 or 4x ≥ 6 x ≤ 0 or x ≥ 3/2

+ 3 + 3

– 5 – 5

+ 3 + 3

Summary

1. Before solving absolute value equations or inequalities, you MUST isolate the absolute value expression and check them are defined or not.

2. Remember telling yourself the jail story. It will help you set up the correct equations or compound inequalities.

3. Solve the equations or compound inequalities and don’t forget the learned knowledge such as when multiplying or dividing a negative number, you need flip the inequality sign.

Solve: 3 1 5x

3 1 5x

5 3 1 5x

4 3 6x

43

2x

Solution Set:

2,

3

42

3

4| orxx

u a u a u a

u a u a u a

i s e q u i v a l e n t t o o r

i s e q u i v a l e n t t o o r

If a is any positive number, then

Theorem

Solve: 4 3 15x

4 3 15x

4 3 15x or 4 315x

4 18x

x 92

4 12x

x 3

),3[2

9,3

2

9|

orxorxx